In this talk I shall discuss Birkhoff-Poritsky conjecture for centrally-symmetric C^2-smooth convex planar billiards. We assume that the domain A between the invariant curve of 4-periodic orbits and the boundary of the phase cylinder is foliated by C^0-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a C^1-smooth foliation by convex caustics of rotation numbers in the interval (0, 1/4] then the boundary curve is an ellipse. The main ingredients of the proof are : <br> (1) the non-standard generating function for convex billiards; <br> (2) the remarkable structure of the invariant curve consisting of 4-periodic orbits; and <br> (3) the integral-geometry approach initiated in [B0], [B1] for rigidity results of circular billiards. <br> Surprisingly, we establish a Hopf-type rigidity for billiards in the ellipse. Based on a joint work with Andrey E. Mironov (Novosibirsk). <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

Several existence and non-existence results for the Nirenberg problem of prescribing the Gauss curvature on a closed surface are classically known. In higher dimension, analog results hold with the Q-curvature replacing the Gauss curvature. In many cases a non-existence result is associated with a blow-up phenomenon which leads to entire solutions of the Liouville equation $-Delta u = e^{2u}$ in dimension 2 or higher-dimensional analogs. On the other hand, Borer, Galimberti and Struwe studied a blow-up phenomenon which could lead to solutions to the equation $$-Delta u =(1-|x|^2)e^{2u} in R^2$$ (1) or $$Delta^2 u =(1-|x|^2)e^{4u} in R^4$$ (2) While non-existence results have been shown for (1) by Struwe, the question remained opened for (2). We recently gave a positive answer with A. Hyder, giving sharp conditions under which (2) admits solutions with controlled behaviour at infinity, hence answering an open question by Struwe. More related open questions will be discussed. <br> <br> <a href="https://teams.microsoft.com/l/meetup-join/19%3a7f273a2aff1145dfae91372d186b1cd9%40thread.tacv2/1612470462653?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%229bfb10cf-6b03-47dc-906c-d23eb368824c%22%7d"> <strong> MS Teams Link for the streaming </strong> <em> </em></a><em> <br> <br> <b> Note</b>: This talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006 </em></div>

A control theory for Hamiltonian systems, based on KAM theory, was introduced in [Ciraolo, 2004] and applied to a model of magnetic field in [Chandre, 2006]. By a combination of Frequency Analysis and of a rigorous (Computer Assisted) KAM algorithm we show that in the phase space of the magnetic field, due to the control term, a set of invariant tori appears, and it acts as a transport barrier. Our analysis, which is common (but often also limited) to celestial mechanics, is very general and can be applied to quasi-integrable Hamiltonian systems satisfying a few additional mild assumptions. <br> <br> <a href="https://teams.microsoft.com/l/meetup-join/19%3a7f273a2aff1145dfae91372d186b1cd9%40thread.tacv2/1611256564820?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22d37d6fea-2e4d-4c35-88e4-99bf4cf68fe9%22%7d"> <strong> MS Teams Link for the streaming </strong> <em> </strong> </a> <br> <strong> Note: </strong> This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

The circulation of our oceans strongly influences climate, weather and biology. Our ocean currents are dynamic, and fluctuate to varying extents. I will introduce data-driven numerical tools that can tease apart dynamic components of the ocean, with information sourced from ocean drifters, satellite imagery, and ocean models. These components, their lifecycles, and their response to external forcing, help us to build a dynamic picture of our ocean. <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

In Algebraic Quantum Field Theory (AQFT), a canonical algebraic construction of the fundamental free field models was provided by Brunetti Guido and Longo in 2002. The Brunetti-Guido-Longo (BGL) construction relies on the identification of spacetime regions called wedges and one-parameter groups of Poincaré symmetries called boosts, the Bisognano-Wichmann property and the CPT-theorem. The last two properties make geometrically meaningful the Tomita-Takesaki theory. In this talk we recall this fundamental structure and explain how the one-particle picture can be generalized. The BGL-construction can start just by considering the Poincaré symmetry group and forgetting about the spacetime. Then it is natural to ask what kind of Lie groups can support a one-particle net and in general a QFT. Given a Z_2-graded Lie group we define a local poset of abstract wedge regions. We provide a classification of the simple Lie algebras supporting abstract wedges in relation with some special wedge configurations. This allows us to exhibit an analog of the Haag-Kastler axioms for one-particle nets undergoing the action of such general Lie groups without referring to any specific spacetime. This set of axioms supports a first quantization net obtained by generalizing the BGL-construction. The construction is possible for a large family of Lie groups and provides several new models. Based on the joint work with Karl-Hermann Neeb (FAU Erlangen-Nürnberg) "Covariant homogeneous nets of standard subspaces" https://arxiv.org/abs/2010.07128 Link al seminario: https://teams.microsoft.com/l/meetup-join/19%3a428ea736adc6424c8ae37f187c91b51b%40thread.tacv2/1611656244968?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22e6325df7-3e74-4c88-ab0b-65ff8a758e69%22%7d

Inspired by the problem of modeling the so called anthropogenic forcing in climatology, e.g. the effects of the emissions of greenhouse gases in the atmosphere, we introduce a novel type of random perturbation for the classical Lorenz flow and prove its stochastic stability. The perturbation acts on the system in an impulsive way, hence is not of diffusive type. Namely, given a cross-section M for the unperturbed flow, each time the trajectory of the system crosses M the phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one. The resulting random evolution is therefore described by a piecewise deterministic Markov process. The proof of the stochastic stability for the unperturbed flow is then carried on working either in the framework of the Random Dynamical Systems or in that of semi-Markov processes. Joint work with Sandro Vaienti. <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

A regular network is a finite union of sufficiently smooth curves whose end points meet in triple junctions. I will present the state-of-the-art of the problem of the motion by curvature of a regular network in the plane mainly focusing on singularity formation. Then I will discuss the need of a “restarting” theorem for networks with multiple junctions of order bigger than three and I will give an idea of a possible strategy to prove it. This is a research in collaboration with Jorge Lira (University of Fortaleza), Rafe Mazzeo (Stanford University) and Mariel Saez (P. Universidad Catolica de Chile). <br> <br> <a href="https://teams.microsoft.com/l/meetup-join/19%3a7f273a2aff1145dfae91372d186b1cd9%40thread.tacv2/1610381100098?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22d37d6fea-2e4d-4c35-88e4-99bf4cf68fe9%22%7d"> <strong> MS Teams Link for the streaming </strong> <em> </strong> </a> <br> <strong> Note: </strong> This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

We prove for the N-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level h>0 of the motion can also be chosen arbitrarily. Our approach is based on the construction of a global viscosity solutions for the Hamilton-Jacobi equation H(x,du(x))=h. Our hyperbolic motion is in fact a calibrating curve of the viscosity solution. The presented results can also be viewed as a new application of Marchal’s theorem, whose main use in recent literature has been to prove the existence of periodic orbits. Joint work with Ezequiel Maderna. <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

The talk is based on joint projects with G. Grillo, K. We prove that the mass conservation property for the heat flow on a complete, connected, noncompact Riemannian manifold $M$, namely stochastic completeness, is equivalent to the uniqueness of nonnegative bounded solutions for a certain class of nonlinear evolution equations. Such a connection was well known in the pure linear case only, i.e. for the heat equation itself. Here we consider equations of the type of $u_t=Delta(phi(u))$, where $phi$ is any nonnegative, concave, increasing function, $C^1$ away from the origin and satisfying $ phi(0)=0 $. We provide optimal criteria for uniqueness/nonuniqueness of nonnegative, bounded (distributional) solutions taking general nonnegative, bounded initial data $u_0$. In particular our results apply to the fast diffusion equation $u_t=Delta(u^m)$ (where $m in (0,1)$), and they show that there is a large class of manifolds in which uniqueness actually fails. This is in sharp contrast, for instance, with the Euclidean case, where existence and uniqueness hold for merely $L^1_{loc}$ initial data thanks to the theory developed by M.A. Herrero and M. Pierre in the '80s. We will also address existence/nonexistence of nonnegative, nontrivial, bounded solutions to a strictly related nonlinear elliptic equation and, if time allows, some work in progress devoted to removing the concavity assumption. <br> The talk is based on joint projects with G. Grillo, K. Ishige and F. Punzo. <br> <br> <a href="https://teams.microsoft.com/l/meetup-join/19%3a7f273a2aff1145dfae91372d186b1cd9%40thread.tacv2/1610040470301?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%229bfb10cf-6b03-47dc-906c-d23eb368824c%22%7d"> <strong> MS Teams Link for the streaming </strong> <em> </em></a><em> <br> <br> <b> Note</b>: This talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006

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